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In [[mathematics]], a number of the expressions that may be encountered in [[calculus]] and occasionally elsewhere are considered to be '''indeterminate forms''', and must be treated as symbolic only, until more careful discussion has taken place. The most common one is
:<math>{0 \over 0}</math>
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The indeterminate form does not imply the limit does not exist. In many cases, algebraic elimination, [[L'Hôpital's rule]], or other methods can be used to simplify the expression so the limit can be more easily evaluated.
==Naive arguments to give indeterminate forms a meaning==
There are many naive reasons which may be given for considering indeterminate forms to have some definite meaning (for example):
*Anything divided by itself is 1. Hence <math>0/0=1</math>
*Anything to the power of 0 is 1 Hence <math>0^0=1</math>
The above are true statements if they are qualified by exceptions. Below are the correct versions of these statements.
*Any non zero number divided by itself is 1.
*Any non zero number to the power of 0 is 1.
The problem with <math>\infty</math> runs deeper. The symbol <math>\infty</math> is not meant to represent a number. It represents a [[limit]] only. As such the following statements are entirely meaningless:
*Anything multiplied by \infty is \infty Hence <math>0 \cdot \infty =\infty</math>
*Anything divided by \infty is 0 Hence <math>\infty /\infty =0</math>
However there are defined concepts such as the [[surreal number]]s and the [[ordinals]] where infinite algebra has been defined.
==Logical circularity==
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